By now you are comfortable with the graphical implications of the first and second derivatives of a function. $f^\prime(x)$ gives you the slope of the tangent while $f^{\prime\prime}(x)$ tells you the concavity of the graph at a point. There is nothing to stop us from finding the third and fourth (and beyond!) derivatives, but there are not any significant graphical characteristics associated with higher derivatives. $f^{\prime\prime\prime}(x)$ is simply the rate of change of $f^{\prime\prime}(x)$ and so on.

1. Find $f^{(4)} (x)$ for $f(x) = x^8$.

   Hint (a):

   Take the derivative four times.

2. If $f(x) = e^{2x}$ , find an expression for the $n$^th derivative, $f^{(n)} (x)$.

   Hint (b):

   Take a few derivatives of this function and look for patterns.